BDD-ALGORITHM

The BDD algorithm in ACL2 uses a combination of manipulation of
IF terms and unconditional rewriting. In this discussion we
begin with some relevant mathematical theory. This is followed by a
description of how ACL2 does BDDs, including concluding discussions
of soundness, completeness, and efficiency.

We recommend that you read the other documentation about BDDs in
ACL2 before reading the rather technical material that follows.
See BDD.

Here is an outline of our presentation. Readers who want a user
perspective, without undue mathematical theory, may wish to skip to
Part (B), referring to Part (A) only on occasion if necessary.

(A) Mathematical Considerations

(A1) BDD term order

(A2) BDD-constructors and BDD terms, and their connection with
aborting the BDD algorithm

(A3) Canonical BDD terms

(A4) A theorem stating the equivalence of provable and syntactic
equality for canonical BDD terms

(B) Algorithmic Considerations

(B1) BDD rules (rules used by the rewriting portion of the ACL2 BDD
algorithm)

(B2) Terms ``known to be Boolean''

(B3) An ``IF-lifting'' operation used by the algorithm, as well as an
iterative version of that operation

(B4) The ACL2 BDD algorithm

(B5) Soundness and Completeness of the ACL2 BDD algorithm

(B6) Efficiency considerations

(A) Mathematical Considerations

(A1) BDD term order

Our BDD algorithm creates a total ``BDD term order'' on ACL2 terms,
on the fly. We use this order in our discussions below of
IF-lifting and of canonical BDD terms, and in the algorithm's use of
commutativity. The particular order is unimportant, except that we
guarantee (for purposes of commutative functions) that constants are
smaller in this order than non-constants.

(A2) BDD-constructors (assumed to be '(cons)) and BDD terms

We take as given a list of function symbols that we call the
``BDD-constructors.'' By default, the only BDD-constructor is
cons, although it is legal to specify any list of function
symbols as the BDD-constructors, either by using the
acl2-defaults-table (see acl2-defaults-table) or by
supplying a :BDD-CONSTRUCTORS hint (see hints). Warning:
this capability is largely untested and may produce undesirable
results. Henceforth, except when explicitly stated to the contrary,
we assume that BDD-constructors is '(cons).

Roughly speaking, a BDD term is the sort of term produced
by our BDD algorithm, namely a tree with all cons nodes lying
above all non-CONS nodes. More formally, a term is said to
be a BDD term if it contains no subterm of either of the
following forms, where f is not CONS.

(f ... (CONS ...) ...)

(f ... 'x ...) ; where (consp x) = t

We will see that whenever the BDD algorithm attempts to create a
term that is not a BDD term, it aborts instead. Thus,
whenever the algorithm completes without aborting, it creates a
BDD term.

(A3) Canonical BDD terms

We can strengthen the notion of ``BDD term'' to a notion of
``canonical BDD term'' by imposing the following additional
requirements, for every subterm of the form (IF x y z):

(a) x is a variable, and it precedes (in the BDD term order)
every variable occurring in y or z;

(b) y and z are syntactically distinct; and,

(c) it is not the case that y is t and z is nil.

We claim that it follows easily from our description of the BDD
algorithm that every term it creates is a canonical BDD term,
assuming that the variables occurring in all such terms are treated
by the algorithm as being Boolean (see (B2) below) and that the
terms contain no function symbols other than IF and CONS.
Thus, under those assumptions the following theorem shows that the
BDD algorithm never creates distinct terms that are provably equal,
a property that is useful for completeness and efficiency (as we
explain in (B5) and (B6) below).

(A4) Provably equal canonical BDD terms are identical

We believe that the following theorem and proof are routine
extensions of a standard result and proof to terms that allow calls
of CONS.

Theorem. Suppose that t1 and t2 are canonical BDD terms
that contain no function symbols other than IF and CONS. Also
suppose that (EQUAL t1 t2) is a theorem. Then t1 and t2
are syntactically identical.

Proof of theorem: By induction on the total number of symbols
occurring in these two terms. First suppose that at least one term
is a variable; without loss of generality let it be t1. We must
prove that t2 is syntactically the same as t1. Now it is
clearly consistent that (EQUAL t1 t2) is false if t2 is a call
of CONS (to see this, simply let t1 be an value that is not a
CONSP). Similarly, t2 cannot be a constant or a variable
other than t1. The remaining possibility to rule out is that
t2 is of the form (IF t3 t4 t5), since by assumption its
function symbol must be IF or CONS and we have already handled
the latter case. Since t2 is canonical, we know that t3 is a
variable. Since (EQUAL t1 t2) is provable, i.e.,

(EQUAL t1 (if t3 t4 t5))

is provable, it follows that we may substitute either t or nil
for t3 into this equality to obtain two new provable equalities.
First, suppose that t1 and t3 are distinct variables. Then
these substitutions show that t1 is provably equal to both t4
and t5 (since t3 does not occur in t4 or t5 by property
(a) above, as t2 is canonical), and hence t4 and t5 are
provably equal to each other, which implies by the inductive
hypothesis that they are the same term -- and this contradicts the
assumption that t2 is canonical (property (b)). Therefore t1
and t3 are the same variable, i.e., the equality displayed above
is actually (EQUAL t1 (if t1 t4 t5)). Substituting t and then
nil for t1 into this provable equality lets us prove (EQUAL t t4)
and (EQUAL nil t5), which by the inductive hypothesis implies
that t4 is (syntactically) the term t and t5 is nil.
That is, t2 is (IF t1 t nil), which contradicts the assumption
that t2 is canonical (property (c)).

Next, suppose that at least one term is a call of IF. Our first
observation is that the other term is also a call of IF. For if
the other is a call of CONS, then they cannot be provably equal,
because the former has no function symbols other than IF and
hence is Boolean when all its variables are assigned Boolean values.
Also, if the other is a constant, then both branches of the IF
term are provably equal to that constant and hence these branches
are syntactically identical by the inductive hypothesis,
contradicting property (b). Hence, we may assume for this case that
both terms are calls of IF; let us write them as follows.

t0: (IF t1 t2 t3)
u0: (IF u1 u2 u3)

Note that t1 and u1 are variables, by property (a) of
canonical BDD terms. First we claim that t1 does not strictly
precede u1 in the BDD term order. For suppose t1 does
strictly precede u1. Then property (a) of canonical BDD terms
guarantees that t1 does not occur in u0. Hence, an argument
much like one used above shows that u0 is provably equal to both
t2 (substituting t for t1) and t3 (substituting nil
for t1), and hence t2 and t3 are provably equal. That
implies that they are identical terms, by the inductive hypothesis,
which then contradicts property (b) for t0. Similarly, u1
does not strictly precede t1 in the BDD term order. Therefore,
t1 and u1 are the same variable. By substituting t for
this variable we see that t2 and u2 are provably equal, and
hence they are equal by the inductive hypothesis. Similarly, by
substituting nil for t1 (and u1) we see that t3 and
u3 are provably, hence syntactically, equal.

We have covered all cases in which at least one term is a variable
or at least one term is a call of IF. If both terms are
constants, then provable and syntactic equality are clearly
equivalent. Finally, then, we may assume that one term is a call of
CONS and the other is a constant or a call of CONS. The
constant case is similar to the CONS case if the constant is a
CONSP, so we omit it; while if the constant is not a CONSP
then it is not provably equal to a call of CONS; in fact it is
provably not equal!

So, we are left with a final case, in which canonical BDD terms
(CONS t1 t2) and (CONS u1 u2) are provably equal, and we want
to show that t1 and u1 are syntactically equal as are t2
and u2. These conclusions are easy consequences of the inductive
hypothesis, since the ACL2 axiom CONS-EQUAL (which you can
inspect using :PE) shows that equality of the given terms is
equivalent to the conjunction of (EQUAL t1 t2) and (EQUAL u1 u2).
Q.E.D.

(B) Algorithmic Considerations

(B1) BDD rules

A rule of class :rewrite (see rule-classes) is said to be
a ``BDD rewrite rule'' if and only if it satisfies the
following criteria. (1) The rule is enabled. (2) Its
equivalence relation is equal. (3) It has no hypotheses.
(4) Its :loop-stopper field is nil, i.e., it is not a
permutative rule. (5) All variables occurring in the rule occur in
its left-hand side (i.e., there are no ``free variables'';
see rewrite). A rule of class :definition
(see rule-classes) is said to be a ``BDD definition rule''
if it satisfies all the criteria above (except (4), which does not
apply), and moreover the top function symbol of the left-hand side
was not recursively (or mutually recursively) defined. Technical
point: Note that this additional criterion is independent of
whether or not the indicated function symbol actually occurs in the
right-hand side of the rule.

Both BDD rewrite rules and BDD definition rules are said to be ``BDD
rules.''

(B2) Terms ''known to be Boolean''

We apply the BDD algorithm in the context of a top-level goal to
prove, namely, the goal at which the :BDD hint is attached. As
we run the BDD algorithm, we allow ourselves to say that a set of
terms is ``known to be Boolean'' if we can verify that the goal
is provable from the assumption that at least one of the terms is
not Boolean. Equivalently, we allow ourselves to say that a set of
terms is ``known to be Boolean'' if we can verify that the original
goal is provably equivalent to the assertion that if all terms in
the set are Boolean, then the goal holds. The notion ``known to be
Boolean'' is conservative in the sense that there are generally sets
of terms for which the above equivalent criteria hold and yet the
sets of terms are not noted as as being ``known to be Boolean.''
However, ACL2 uses a number of tricks, including type-set
reasoning and analysis of the structure of the top-level goal, to
attempt to establish that a sufficiently inclusive set of terms is
known to be Boolean.

From a practical standpoint, the algorithm determines a set of terms
known to be Boolean; we allow ourselves to say that each term in
this set is ``known to be Boolean.'' The algorithm assumes that
these terms are indeed Boolean, and can make use of that assumption.
For example, if t1 is known to be Boolean then the algorithm
simplifies (IF t1 t nil) to t1; see (iv) in the discussion
immediately below.

(B3) IF-lifting and the IF-lifting-for-IF loop

Suppose that one has a term of the form (f ... (IF test x y) ...),
where f is a function symbol other than CONS. Then we say
that ``IF-lifting'' test ``from'' this term produces the
following term, which is provably equal to the given term.

Here, we replace each argument of f of the form (IF test .. ..),
for the same test, in the same way. In this case we say that
``IF-lifting applies to'' the given term, ``yielding the test''
test and with the ``resulting two branches'' displayed above.
Whenever we apply IF-lifting, we do so for the available test
that is least in the BDD term order (see (A1) above).

We consider arguments v of f that are ``known to be Boolean''
(see above) to be replaced by (IF v t nil) for the purposes of
IF-lifting, i.e., before IF-lifting is applied.

There is one special case, however, for IF-lifting. Suppose that
the given term is of the form (IF v y z) where v is a variable
and is the test to be lifted out (i.e., it is least in the BDD term
order among the potential tests). Moroever, suppose that neither
y nor z is of the form (IF v W1 W2) for that same v.
Then IF-lifting does not apply to the given term.

We may now describe the IF-lifting-for-IF loop, which applies to
terms of the form (IF test tbr fbr) where the algorithm has
already produced test, tbr, and fbr. First, if test is
nil then we return fbr, while if test is a non-nil
constant or a call of CONS then we return tbr. Otherwise, we
see if IF-lifting applies. If IF-lifting does not apply, then we
return (IF test tbr fbr). Otherwise, we apply IF-lifting to
obtain a term of the form (IF x y z), by lifting out the
appropriate test. Now we recursively apply the IF-lifting-for-IF
loop to the term (IF x y z), unless any of the following special
cases apply.

(i) If y and z are the same term, then return y.

(ii) Otherwise, if x and z are the same term, then replace
z by nil before recursively applying IF-lifting-for-IF.

(iii) Otherwise, if x and y are the same term and y is
known to be Boolean, then replace y by t before recursively
applying IF-lifting-for-IF.

(iv) If z is nil and either x and y are the same term or
x is ``known to be Boolean'' and y is t, then return x.

NOTE: When a variable x is known to be Boolean, it is easy to
see that the form (IF x t nil) is always reduced to x by this
algorithm.

(B4) The ACL2 BDD algorithm

We are now ready to present the BDD algorithm for ACL2. It is given
an ACL2 term, x, as well as an association list va that
maps variables to terms, including all variables occurring in x.
We maintain the invariant that whenever a variable is mapped by
va to a term, that term has already been constructed by the
algorithm, except: initially va maps every variable occurring in
the top-level term to itself. The algorithm proceeds as follows.
We implicitly ordain that whenever the BDD algorithm attempts to
create a term that is not a BDD term (as defined above in
(A2)), it aborts instead. Thus, whenever the algorithm completes
without aborting, it creates a BDD term.

If x is a variable, return the result of looking it up in va.

If x is a constant, return x.

If x is of the form (IF test tbr fbr), then first run the
algorithm on test with the given va to obtain test'. If
test' is nil, then return the result fbr' of running the
algorithm on fbr with the given va. If test' is a constant
other than nil, or is a call of CONS, then return the result
tbr' of running the algorithm on tbr with the given va. If
tbr is identical to fbr, return tbr. Otherwise, return the
result of applying the IF-lifting-for-IF loop (described above) to
the term (IF test' tbr' fbr').

If x is of the form (IF* test tbr fbr), then compute the
result exactly as though IF were used rather than IF*, except
that if test' is not a constant or a call of CONS (see
paragraph above), then abort the BDD computation. Informally, the
tests of IF* terms are expected to ``resolve.'' NOTE: This
description shows how IF* can be used to implement conditional
rewriting in the BDD algorithm.

If x is a LAMBDA expression ((LAMBDA vars body) . args)
(which often corresponds to a LET term; see let), then first
form an alist va' by binding each v in vars to the result
of running the algorithm on the corresponding member of args,
with the current alist va. Then, return the result of the
algorithm on body in the alist va'.

Otherwise, x is of the form (f x1 x2 ... xn), where f is a
function symbol other than IF or IF*. In that case, let
xi' be the result of running the algorithm on xi, for i
from 1 to n, using the given alist va. First there are a few
special cases. If f is EQUAL then we return t if x1' is
syntactically identical to x2' (where this test is very fast; see
(B6) below); we return x1' if it is known to be Boolean and
x2' is t; and similarly, we return x2' if it is known to be
Boolean and x1' is t. Next, if each xi' is a constant and
the :executable-counterpart of f is enabled, then the
result is obtained by computation. Next, if f is BOOLEANP and
x1' is known to be Boolean, t is returned. Otherwise, we
proceed as follows, first possibly swapping the arguments if they
are out of (the BDD term) order and if f is known to be
commutative (see below). If a BDD rewrite rule (as defined above)
matches the term (f x1'... xn'), then the most recently stored
such rule is applied. If there is no such match and f is a
BDD-constructor, then we return (f x1'... xn'). Otherwise, if a
BDD definition rule matches this term, then the most recently stored
such rule (which will usually be the original definition for most
users) is applied. If none of the above applies and neither does
IF-lifting, then we return (f x1'... xn'). Otherwise we apply
IF-lifting to (f x1'... xn') to obtain a term (IF test tbr fbr);
but we aren't done yet. Rather, we run the BDD algorithm (using the
same alist) on tbr and fbr to obtain terms tbr' and
fbr', and we return (IF test tbr' fbr') unless tbr' is
syntactically identical to fbr', in which case we return tbr'.

When is it the case that, as said above, ``f is known to be
commutative''? This happens when an enabled rewrite rule is of the
form (EQUAL (f X Y) (f Y X)). Regarding swapping the arguments
in that case: recall that we may assume very little about the BDD
term order, essentially only that we swap the two arguments when the
second is a constant and the first is not, for example, in (+ x 1).
Other than that situation, one cannot expect to predict accurately
when the arguments of commutative operators will be swapped.

(B5) Soundness and Completeness of the ACL2 BDD algorithm

Roughly speaking, ``soundness'' means that the BDD algorithm should
give correct answers, and ``completeness'' means that it should be
powerful enough to prove all true facts. Let us make the soundness
claim a little more precise, and then we'll address completeness
under suitable hypotheses.

Claim (Soundness). If the ACL2 BDD algorithm runs to
completion on an input term t0, then it produces a result that is
provably equal to t0.

We leave the proof of this claim to the reader. The basic idea is
simply to check that each step of the algorithm preserves the
meaning of the term under the bindings in the given alist.

Let us start our discussion of completeness by recalling the theorem
proved above in (A4).

Theorem. Suppose that t1 and t2 are canonical BDD terms
that contain no function symbols other than IF and CONS. Also
suppose that (EQUAL t1 t2) is a theorem. Then t1 and t2
are syntactically identical.

Below we show how this theorem implies the following completeness
property of the ACL2 BDD algorithm. We continue to assume that
CONS is the only BDD-constructor.

Claim (Completeness). Suppose that t1 and t2 are
provably equal terms, under the assumption that all their variables
are known to be Boolean. Assume further that under this same
assumption, top-level runs of the ACL2 BDD algorithm on these terms
return terms that contain only the function symbols IF and
CONS. Then the algorithm returns the same term for both t1
and t2, and the algorithm reduces (EQUAL t1 t2) to t.

Why is this claim true? First, notice that the second part of the
conclusion follows immediately from the first, by definition of the
algorithm. Next, notice that the terms u1 and u2 obtained by
running the algorithm on t1 and t2, respectively, are provably
equal to t1 and t2, respectively, by the Soundness Claim. It
follows that u1 and u2 are provably equal to each other.
Since these terms contain no function symbols other than IF or
CONS, by hypothesis, the Claim now follows from the Theorem above
together with the following lemma.

Lemma. Suppose that the result of running the ACL2 BDD
algorithm on a top-level term t0 is a term u0 that contains
only the function symbols IF and CONS, where all variables of
t0 are known to be Boolean. Then u0 is a canonical BDD term.

Proof: left to the reader. Simply follow the definition of the
algorithm, with a separate argument for the IF-lifting-for-IF loop.

Finally, let us remark on the assumptions of the Completeness Claim
above. The assumption that all variables are known to be Boolean is
often true; in fact, the system uses the forward-chaining rule
boolean-listp-forward (you can see it using :pe) to try to
establish this assumption, if your theorem has a form such as the
following.

Moreover, the :BDD hint can be used to force the prover to abort
if it cannot check that the indicated variables are known to be
Boolean; see hints.

Finally, consider the effect in practice of the assumption that the
terms resulting from application of the algorithm contain calls of
IF and CONS only. Typical use of BDDs in ACL2 takes place in
a theory (see theories) in which all relevant non-recursive
function symbols are enabled and all recursive function symbols
possess enabled BDD rewrite rules that tell them how open up. For
example, such a rule may say how to expand on a given function
call's argument that has the form (CONS a x), while another may
say how to expand when that argument is nil). (See for example
the rules append-cons and append-nil in the documentation for
IF*.) We leave it to future work to formulate a theorem that
guarantees that the BDD algorithm produces terms containing calls
only of IF and CONS assuming a suitably ``complete''
collection of rewrite rules.

(B6) Efficiency considerations

Following Bryant's algorithm, we use a graph representation of
terms created by the BDD algorithm's computation. This
representation enjoys some important properties.

(Time efficiency) The test for syntactic equality of BDD terms is
very fast.

Implementation note. The representation actually uses a sort
of hash table for BDD terms that is implemented as an ACL2
1-dimensional array. See arrays. In addition, we use a second
such hash table to avoid recomputing the result of applying a
function symbol to the result of running the algorithm on its
arguments. We believe that these uses of hash tables are standard.
They are also discussed in Moore's paper on BDDs; see bdd for
the reference.